MASSACHUSETTS INSTITUTE OF TECHNOLOGY
16.888/ESD.77 MSDO
16.888/ESD.77J Multidisciplinary System Design Optimization (MSDO)
Spring 2010
Assignment 4
Instructors:
Prof. Olivier de Weck
Prof. Karen Willcox
Dr. Anas Alfaris
Dr. Douglas Allaire
TAs:
Andrew March
Issued:
Kaushik Sinha
Lecture 13
Due:
Lecture 17
You are expected to solve Part (a) individually and Part (b) in your project team. Each person
must submit their own Part (a) but you should submit Part (b) as a group. Please indicate the
name(s) of your teammate(s).
Topics: genetic algorithms, mixed-integer optimization, scaling
Part (a)
Part A1
The following questions refer to a Genetic Algorithm as defined in Lecture 10:
a) Which of the following will most increase population diversity?
a. Increasing mutation rate
b. Changing the crossover location
c. Increasing the amount of elitism
d. All of the above
e. None of the above
b) In a binary GA, how many bits are needed to represent numbers between 1 and 10 with a
resolution of 0.0001?
a. 3
b. 4
c. 16
d. 17
e. 18
c) In a binary GA, what is the minimum number of bits required to represent a discrete
design variable that can have values, 1, 2, 3 or 4?
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
16.888/ESD.77 MSDO
a.
b.
c.
d.
1
2
3
4
d) For a discrete variable that can have the values, 1, 2, 3, or 4, in the binary representation
of this variable with the minimum number of bits, what would the representation of 3 be?
a. 1
b. 01
c. 10
d. 111
e. 010
f. 0110
e) Given the parents 01001101 and 01100100, which of the following are possible single
point crossovers?
A: 11001101 B: 01000100 C: 01001100 D: 01111101
a. A & B
b. B & C
c. A & D
d. A & C
e. A, B, C & D
f) In a population, the fitness function values of each member are: F1=100, F2=800, F3=1,
F4=90, F5=9. Using roulette wheel selection, what is the approximate probability the
member with F1 will be chosen as a parent?
a. 0.01
b. 0.10
c. 0.80
d. 0.90
e. 0.99
g) In a population, the fitness function values of each member are: F1=100, F2=800, F3=1,
F4=90, F5=9. Now, using the fraction of the population a member dominates as the
selection criterion, what is the approximate probability the member with F1 will be
chosen as a parent?
a. 1
b. 0.90
c. 0.80
d. 0.50
e. 0.10
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY 16.888/ESD.77 MSDO h) In a binary encoded GA with two design variables, one with 4 bits, and one with 16 bits,
what is the total number of possible population members?
a. 218
b. 220
c. 224
d. 232
Part A2
Your objective is to design the cheapest possible bridge to span a highway. The total span of the
bridge (across both halves of the highway) must be L=30 meters, and it must support its own
weight and a load q=33x104 N/m along its span (a total load of about 1x106 N plus its own
weight).
I-beams
Middle Support
The bridge span will be supported by between one and four I-beams. In the figure above, the Ibeams would be parallel to each other going into the page, for example it could be one I-beam in
the middle of the bridge, or one I-beam on both sides of the bridge, etc. and this will be
represented by the design variable, nIbeams. The shape of the I-beams will be represented by three
continuous design variables, the height, h, flange width, b, and thickness, t. The middle support
will be rectangular (when viewed from above), and will have two design variables, the width, w,
and depth, d.
b
w
t
d
h
Where, ρIbeams, is the density of the material used for the I-beams, the mass of the I-beams can be
computed using,
M Ibeams = [2bt + (h − 2t )t ]Lρ Ibeams n Ibeams ,
and where, ρSupport , is the density of the material used for the support, and H=5m is the height of
the bridge above the ground, the mass of the middle support can be computed using,
M Support = wdHρ Support .
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY 16.888/ESD.77 MSDO There is a constraint that the stress the I-beam is less than the material failure stress for the Ibeam, σFailure-Ibeams. Note: g is the gravitational constant (9.81 m/s2).
2
⎛ L⎞
⎛ L⎞
q⎜ ⎟ + M Ibeams ⎜ ⎟ g
2
⎝ 4⎠ ⎛h⎞ ≤σ
σ Ibeams = ⎝ ⎠
⎜ ⎟
Failure− Ibeams
8 I Ibeam n Ibeams
⎝2⎠
Where, IIbeam, is the moment of inertia for the I-beam given by,
2
3
⎡ t 3b
(
h − 2t ) t
⎛h t ⎞ ⎤
+ tb⎜ − ⎟ ⎥ .
+ 2⎢
I Ibeam =
12
⎝ 2 2 ⎠ ⎥⎦
⎢⎣ 12
In addition that the shear stress in the I-beams is less than the material failure stress:
M Ibeams g + qL
τ Ibeams =
≤ σ Failure−Ibeams .
4[2bt + (h − 2t )t ]n Ibeams
For the middle-support there are two constraints, the column cannot buckle and the stress must
be less than the material failure stress. Buckling is based on a requirement that the applied load is
less than a critical load,
M
g + qL
PApplied = Ibeams
≤ PCrit
2
Where the critical load is a function of the lowest moment of inertia of the support and the
modulus of elasticity of the support material, Esupport,
⎧ w 3d wd 3 ⎫
2
,
π E Support min ⎨
⎬
12 12 ⎭
⎩
PCrit =
.
4H 2
The stress requirement is that the applied stress is less than the support material failure stress,
P
σ Support = Applied ≤ σ Failure− Support
wd
The bridge span (I-beams) can be made from Al 6061, A36 Steel, A514 Steel, or Titanium;
however, the support can be made from Al 6061, A36 Steel, A514 Steel, or Concrete. The reason
for the difference is that concrete cannot be loaded in tension. The material properties and prices
are listed in the Table:
Material
Al 6061
A36 Steel
A514 Steel
Titanium
Concrete
Density
(kg/m3)
2700
7850
7900
4500
2400
Modulus of Elasticity
(GPa=109N/m2)
70
210
210
120
31
Failure Stress
(MPa=106N/m2)
270
250
700
760
70
Cost ($/kg)
2.05
0.62
0.90
16.00
0.04
Your objective is to find the dimensions of the I-beams, number of I-beams, and material type
for the I-beams, as well as the dimensions of the support and material type for the support to
minimize cost of the bridge. Where cIbeams, and cSupport are the cost per kilogram of the materials
used for the I-beams and Support, the total bridge cost is:
C = c Ibeams M Ibeams + M Support cSupport
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
16.888/ESD.77 MSDO
a) Please explain what optimization algorithm you chose to find the cheapest possible
bridge and why.
b) What are the design variables and cost for the cheapest bridge?
Part A3
Consider the 8-dimensional function:
1
f (x) = (0.0001x12 + 0.001x 22 + 0.01x32 + 0.1x 42 + x52 + 10x62 + 100x72 + 1000x82 )
2
Note: for this question you may either create your own optimization algorithms or use any
optimization package you like that has the required capabilities.
a) Minimize this function using Newton’s method from 10 random starting points on the
interval xi ∈ [− 5,5] and consider convergence when ∇f (x ) ≤ 10 −8 . What is the average
number of iterations required for the optimization to converge?
b) Repeat the optimization in part (a) this time using either a conjugate gradient method or a
quasi-Newton method. If you use a quasi-Newton method ensure the initial Hessian
estimate is the identity matrix. On average how many iterations are required for the
optimization to converge?
c) Repeat the optimization in part (a) this time using steepest descent. On average how
many iterations are required for the optimization to converge?
d) What is the condition number of the Hessian?
e) Find a non-singular transformation x = Ly such that the condition number of the Hessian
is 1.
f) Repeat parts a, b, and c on the scaled problem. What are the average number of iterations
each method requires to minimize f (x) ?
g) What can you say about the sensitivity of each of these optimization algorithms to
scaling?
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY 16.888/ESD.77 MSDO Part (b)
(b1) Scaling
Use a gradient-based algorithm as in A3 for the question below (pick one algorithm). The current
optimal solution x* refers to the “optimal” solution in your project found using this algorithm in
A3.
(b1.1) Compute the n diagonal entries of the Hessian at your current optimal solution, H(x*).
Use finite differencing to evaluate these entries. (Note: if you have a large number of design
variables, you may do these steps for any 10 design variables.)
(b1.2) If any of the entries computed in (b1.1) are greater than 102 or less than 10-2, then the
corresponding design variable should be scaled. Note that Hii ~1/xi2, so if Hii ~ 10-4 the
appropriate scaling is Xi=10-2xi. Compute the scaling required for each design variable to make
Hii ~ O(1). Consider only scalings of the form 10-2, 10-1, 101, 102 etc. (i.e. worry about magnitude
only).
(b1.3) Redefine your design variables using the scalings computed in (b1.2) and re-run the
optimizer, starting from the previous “optimal” solution x*. Do you see any change in the
optimal solution?
Please note: If you do not have any continuous design variables in your problem, please instead
analyze the scaling of your constraints. Study whether scaling certain constraints in your problem
affects convergence or convergence rates.
(b2) Heuristic Optimization
In A3 you used a gradient search technique to optimize the system for your design project. In this
assignment we want you to apply a heuristic technique (SA, GA, PSO or Tabu Search) to your
design project.
(b2.1) Describe which technique you chose and why.
(b2.2) Attempt to optimize the system and compare the answers with the answers you received
with the gradient search technique.
(b2.3) Tune the parameters of the algorithm and observe the differences in behavior. What
appears to be the best algorithm tuning parameter settings for your project?
(b2.4) How confident are you that you have found the true global optimum?
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ESD.77 / 16.888 Multidisciplinary System Design Optimization
Spring 2010
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